CAS MAS-IIModern Actuarial Statistics II

For a class of risks, the expected process variance is 200 and the variance of hypothetical means is 50. A risk has 8 years of experience. Calculate the Bühlmann credibility factor.

Annual claim counts are Poisson($\lambda$). The prior for $\lambda$ is gamma with $\alpha=3$, $\theta=0.5$. A risk has 4 claims in 2 years. Calculate the posterior mean of $\lambda$.

Using limited-fluctuation credibility, the standard for full credibility is that the observed claim count be within 5% of expected with probability 90%. Claim counts are Poisson. Calculate the number of claims needed for full credibility.

A Metropolis sampler with a symmetric proposal is at state $x$ with target density $\pi(x)=0.50$. The proposed state $y$ has $\pi(y)=0.20$. Calculate the probability the proposal is accepted.

Which statement about regularized regression is correct?

Three policyholders have the following exposures and claim counts over the same period: A: 10 exposures, 3 claims; B: 30 exposures, 6 claims; C: 60 exposures, 9 claims. Assume Poisson frequency per exposure with $EPV=0.20$ and $VHM=0.01$. Compute the Bühlmann–Straub credibility-weighted frequency for each policyholder.

A linear mixed model for territory-level pure premium includes a fixed intercept and a random territory intercept $b_i\sim N(0,\tau^2)$ with residual variance $\sigma^2$. If $\tau^2=50$ and $\sigma^2=200$, calculate the intraclass correlation (the correlation between two observations in the same territory).

Full credibility for claim frequency requires 1,082 claims (from the usual $z_{0.95}=1.645$, $r=5\%$ standard). If claim severity has a coefficient of variation of 2.0, calculate the full credibility standard (in claim counts) for aggregate losses.

A claim severity mean $\mu$ has a Normal(100, 400) prior. A sample of 5 claims (assume known variance 900 per claim) has sample mean 130. Calculate the Bayesian posterior mean of $\mu$.

In gradient boosting for a Poisson loss model, decreasing the learning rate (shrinkage) while holding the number of trees fixed typically:

A workers' compensation pool has three insureds with the following exposures and claim counts over 3 years: Insured 1: 20 exposures, 5 claims; Insured 2: 50 exposures, 8 claims; Insured 3: 100 exposures, 12 claims. The pool's overall EPV is 0.15 and VHM is 0.005. (a) Compute Bühlmann–Straub credibility for each insured. (b) Compute each insured's credibility-weighted estimated frequency using the pool average as the complement of credibility. (c) Explain qualitatively how the estimates would change if VHM were larger.